Question: We know the following to be true:

$\bullet$ 1.  $Z$ and $K$ are integers with $500 < Z < 1000$ and $K > 1;$

$\bullet$ 2.  $Z$ = $K \times K^2.$

What is the value of $K$ for which $Z$ is a perfect square?
Explanation: From the second fact, we know that $Z=K^3.$ $Z$ is a perfect square if $K^3$ is a perfect square, so $Z$ is the sixth power of some integer. Since $500<Z<1000,$ the only value of $Z$ that works is $Z=3^6=729.$ Thus, $K=\sqrt[3]{729}=\boxed{9}.$